I was last at BSL in 2022 and you could "freely" cross the customs border landside on the lowest level of the terminal. There was a desk but it was unstaffed. I went back and forth 4 times.

Although bus 50 is obviously more frequent and convenient to get to the SBB station, from the French side you can take bus 11 to St Louis station followed by a tram or a train into Switzerland if you wish.